Question

Prove that the sum of n arithmetic means between two numbers is n times the single arithmetic mean between them.

Answer 1

Let n a.m, between a,b are a1,a2,a3,............an a,a1,a2,a3,.............an,b sum of n a.m. Sn = n[2a1 + (n-1)d]/2 ----------(1) a.m. of a,b is (a+b)/2 b = a+(n+1)d a.m. = [2a+(n+1)d]/2 -------------(2) a1 = a+d put in (1) Sn = n[2a + 2d + (n-1)d]/2 = n[2a+(n+1)d]/2 ------------(3) from equation (2) & (3) Sn = n*a.m. where d = common difference of a.p.